Higher-order finite element methods for the nonlinear Helmholtz equation
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In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and (pre-asymptotic) error estimates of the finite element solution under a resolution condition between the wave number k, the mesh size h and the polynomial degree p of the form “k(kh)^p sufficiently small” and a so-called smallness of the data assumption. For the latter, we prove that the logarithmic dependence in h from the case p=1 in [H. Wu, J. Zou, SIAM J. Numer. Anal. 56(3): 1338-1359, 2018] can be removed for p≥ 2. We show convergence of two different fixed-point iteration schemes. Numerical experiments illustrate our theoretical results and compare the robustness of the iteration schemes with respect to the size of the nonlinearity and the right-hand side data.
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